We start by noting that \(\hat{\phi }^c\)  is also \(c\) -concave by definition, and Lemma 2 in [1]} implies that \(\hat{\phi }^c\)  is \(|{\mathcal {M}}|\) -Lipschitz, namely
\(\left|\hat{\phi }^c(x)-\hat{\phi }^c(y)\right|\le |{\mathcal {M}}| d(x,y),\) 
